Tuesday, June 10, 2008

some thoughts on the so-called “liar's paradox” and a riddle.

Consider the following sentence:

The second sentence of this post is false. The previous sentence is said to engender a paradox because if the sentence is true, i.e. that it is true that the second sentence is false, then it also entails that the sentence cannot be true, for the sentence says of itself that it is false. Conversely, if the sentence is false, i.e. if the second sentence is false in saying that it is false, then the sentence is expressing a truth; but this is impossible because the sentence says that it is false. Nay, paradox is not avoided on either reading.

It has been said that we can avoid the paradox by showing that the second sentence cannot have a truth value in the first place because it fails to intend anything other than it self, i.e. that it does not refer to anything beyond itself. Consider the following sentence:

My bookshelf is false.

We think that such a sentence is non-sense because bookshelves are not the sort of things that can possibly be true or false and this is because bookshelves lack the power of intentionality, they do not ever refer to things. Sentences, on the other hand, do refer to things in virtue of their propositional content. Consider the following sentence:

My bookshelf is shorter than the ceiling.

This sentence refers to something beyond itself and therefore has the ability to be true or false. So what of the second sentence? Isn’t it a sentence and thereby eligible for truthvaluehood? True, the second sentence is a sentence, but because it refers to itself alone and not beyond itself it loses the power to intend, i.e. in failing to refer beyond itself it becomes an object like my bookshelf, and therefore it cannot have a truth value. And so the paradox is solved by showing that purely self-referential sentences cannot be true and false, and therefore are not paradox-eligible.

So what of the famous Cretan who said “all Cretans are liars”? Admittedly, such a Cretan would be expressing a paradox so long as we understand him to mean “all” in an unqualified sense. For if all Cretans are liars and this particular Cretan says “all Cretans are liars” then it means not all Cretans are liars, for this particular Cretan is telling the truth, in which case it would be false that all Cretans are liars, in which case his sentence would be false. The only way out of this paradox, it seems to me, is to say that if it is true that all Cretans are liars, then no Cretan should be able to say so.

And now I’ll end with a riddle. You’re on you way down a mountain and you come to a fork in the road in which there is two paths to take. You know that one path will bring you to your destruction and the other to safety, but you don’t know which. Lo and behold there are twin brothers living in a cabin at the fork in the road and you know that one always tells the truth and the other always lies, and you don't know who is who. What single question could you ask them both that will let you know for certain which road leads where?


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